"To publish the emails" means to make the emails available or share them with others, usually by making them public or putting them online for everyone to see and read. Full definition
We won't publish email addresses in the print edition. (businessnews.com.au)
Therefore, I decided a more interesting way to discuss the topic was to publish an email exchange I had with a friend and fellow absolute return investor. (ericcinnamond.com)
Whoever published these emails likely wanted to spread misinformation about climate science to try to undermine the conference. (ucsusa.org)